Ratner's property and mixing for special flows over two-dimensional rotations

We consider special flows over two-dimensional rotations by $(\alpha,\beta)$ on $\T^2$ and under piecewise $C^2$ roof functions $f$ satisfying von Neumann's condition $\int_{\T^2}f_x(x,y)\,dx\,dy\neq 0\neq \int_{\T^2}f_y(x,y)\,dx\,dy.$ Such flows are shown to be always weakly mixing and never partially rigid. For an uncountable set of $(\alpha,\beta)$ with both $\alpha$ and $\beta$ of unbounded partial quotients the strong mixing property is proved to hold. It is also proved that while specifying to a subclass of roof functions and to ergodic rotations for which $\alpha$ and $\beta$ are of bounded partial quotients the corresponding special flows enjoy so called weak Ratner's property. As a consequence, such flows turn out to be mildly mixing.

It is already in 1932 when von Neumann [25] considered special flows over irrational rotations on T = [0, 1) under roof functions f which were piecewise C 1 . He proved weak mixing of such flows whenever the condition was satisfied. Linear functions f (x) = ax + b for 0 ≤ x < 1 (with a = 0 and b ∈ R so that f > 0) are the simplest examples of roof functions satisfying von Neumann's condition (1). Piecewise C 1 -functions are of bounded variation, hence, as shown by Kochergin [18] in 1972, the corresponding special flows are not mixing. A natural question whether a special flow over an irrational rotation by α ∈ [0, 1) under f piecewise C 1 and satisfying (1) can enjoy a stronger property than weak mixing was answered positively in [8]; indeed, such flows turn out to be mildly mixing whenever α has bounded partial quotients. As a matter of fact, the mild mixing property has been proved in [8] in two independent steps: first, the absence of partial rigidity (which does not require any Diophantine condition on α) has been proved and then so called Ratner's property has been established for α with bounded partial quotients.
In the present paper we consider special flows over an egodic two-dimensional rotation T (x, y) = (x + α, y + β). Our roof functions f : T 2 → R + will be piecewise C 2 (discontinuities of f are contained in finitely many horizontal and vertical lines, see Definition 3) and will satisfy a two-dimensional analog of (1) (2)  In what follows (2) will be referred to as the weak von-Neumann's condition. We will observe that this condition implies the weak mixing property of the corresponding special flows T f (Theorem 3.2) as well as the absence of partial rigidity (Theorem 4.1). As in [8], our aim will be to prove that such flows are mildly mixing. If we want the strategy from [8] of showing the mild mixing property (under some Diophantine assumptions on (α, β)) to work we need to prove an analog of Ratner's property for such flows. This is done only partially, namely, in a restricted class of roof functions satisfying (2) and both α and β are assumed to have bounded partial quotients, see Theorem 7.4 in which so called weak Ratner's property is proved to hold. The class of roof functions includes all positive linear functions f (x, y) = ax + by + c with a/b ∈ R \ Q. Then, the mild mixing property follows (Theorem 8.2). Proving (even the weak) Ratner's property of such flows is of independent interest, as it has some other ergodic consequences (Theorem 5.9, see also [28]). Recall that the original notion, introduced by Ratner in [26] and called there H p -property, is as follows: Ratner's property. Let (X, d) be a σ-compact metric space, µ a probability Borel measure on (X, d) and (S t ) t∈R a µ-preserving flow. The flow (S t ) t∈R is called H pflow, p = 0, if for every ε > 0 and N ∈ N there exist κ = κ(ε) > 0, δ = δ(ε, N ) > 0 and a Borel subset Z = Z(ε, N ) ⊂ X with µ(Z) > 1 − ε such that if x, x ′ ∈ Z, x ′ is not in the orbit of x and d(x, x ′ ) < δ, then there are M = M (x, x ′ ) ≥ N , L = L(x, x ′ ) ≥ N with L/M ≥ κ such that if we denote K ± = {n ∈ Z ∩ [M, M + L] : d(S np (x), S (n±1)p (x ′ )) < ε} then either #K + /L > 1 − ε or #K − /L > 1 − ε.
Ratner's property, originally proved by M. Ratner [26] for horocycle flows, in the framework of special flows over irrational rotations first appeared in [8]. In fact, already in [8] the original definition of Ratner has been modified and ±p was replaced by a finite subset of R \ {0}. In the present paper we need a further weakening of the definition: we introduce a compact set P ⊂ R \ {0} so that the orbits of two close different points are close up to a shift of time belonging to P on sufficiently long pieces of orbits. We call this property weak Ratner's property (see Definition 4).
Unlike the one-dimensional rotation case, special flows over two-dimensional rotations even under smooth functions can be mixing, see [5], [6]. In Section 9 we show that special flows with piecewise C 2 roof functions and satisfying the following strong von Neumann's condition (3) T 2 f x (x, y) dx dy = 0 and T 2 f y (x, y) dx dy = 0 are mixing for uncountably many (α, β) ∈ T 2 (Theorem 9.3). The main tool to prove mixing property we use is a Fayad's criterion from [5]. In particular, in the linear case f (x, y) = ax + by + c mixing is possible for a special choice of α, β -a phenomenon which can not happen in the one-dimensional case.
1.1. Plan of the paper. The plan on the paper is as follows: Section 2 introduces terminology and notation that will be used throughout the remainder of the paper. In Section 3 we will show weak mixing of the special flow T f (Theorem 3.2) assuming that the roof function f : T 2 → R + is piecewise C 2 and satisfies (2). In Section 4 we will establish the absence of partial rigidity under the same assumption (Theorem 4.1). The proofs of these results are proved in spirit to the one-dimensional case in [8].
Next part of the paper deals with mild mixing. We use a criterion from [8]: If a flow is not partially rigid and it is a finite extension of each of its non-trivial factors (finite fibers factor property) then it is mildly mixing. The absence of partial rigidity being already established, in order to deal with the second assumption the notion of weak Ratner's property is introduced in Section 5. Then, in Theorem 5.9, it is proved that weak Ratner's property implies finite fibers factor property.
In Section 6 we present techniques (Lemma 6.3 and Proposition 6.4) that help us in proving the weak Ratner property to hold for special flows built over rotations. In Section 7 we introduce a class of piecewise C 2 von Neumann roof functions on T 2 and we consider the corresponding special flows over ergodic rotations whose both coordinates have bounded partial quotients. Using techniques from Section 6 for this class of special flows, we prove weak Ratner's property (see Theorem 7.4), which finally establishes mild mixing. Moreover, in Section 8 we provide an example from this class which is mildly mixing but is not mixing. Section 9 deals with mixing property for special flows with piecewise C 2 roof functions satisfying strong von Neumann's condition (3) and it uses methods different from earlier sections. We first notice that Fayad's criterion [5] (alternating uniform stretch of the Birkhoff sums in the vertical and horizontal directions) of mixing of special flows for C 2 roof functions can be extended to piecewise C 2 case. Then we prove mixing over an uncountable family of rotations by (α, β) on T 2 (both α and β have unbounded partial quotients).
We will discuss some other consequences of the results proved in the paper as well as some open problems in Section 10.
Our special thanks go to A. Katok who was the first to conjecture that already linearity over two dimensional rotations may be sufficient for strong mixing property of the corresponding special flows. Such mixing flows are apparently the simplest examples of mixing flows in the framework of special flows under regular roof functions and over multi-dimensional rotations.
We also thank both referees for numerous comments and suggestions which led both to a better presentation as well as to stronger results than in the first version of the paper. Especially, we thank one of the referees for proposing the main idea of the proof of Theorem 7.4.

Notation
Let T be an ergodic automorphism of a standard probability Borel space (X, B, µ), this is for every T -invariant set A ∈ B, either A or its complement X \ A has measure zero. Assume f : X → R is a strictly positive integrable function and let B(R) and λ R denote Borel σ-algebra and Lebesgue measure on R respectively. Then by T f = (T f t ) t∈R we will mean the corresponding special flow under f (see e.g. [3], Under the action of the flow T f each point in X f moves vertically at unit speed, and we identify the point Then for every (x, s) ∈ X f we have If X is equipped with a metric d whose Borel σ-algebra is equal to B then we will consider on X f the metric d f defined by then (S t ) t∈R is weakly mixing.
Of course, mixing implies weak mixing, and the following conditions are equivalent (see [3]): (i) (S t ) t∈R is weakly mixing; (ii) the Cartesian product flow (S t × S ′ t ) t∈R is ergodic provided that (S ′ t ) t∈R is an ergodic flow on a standard probability Borel space; (iii) if F : X → C is an eigenfunction corresponding to an eigenvalue θ ∈ R, i.e.
F (S t x) = e itθ F (x) then θ = 0 and F is constant.
Definition 2. A measure-preserving flow (S t ) t∈R on a standard probability Borel space is mildly mixing if its Cartesian product with an arbitrary ergodic (finite or infinite conservative) measure-preserving transformation remains ergodic.
Recall that a measure-preserving flow (S ′ t ) t∈R on a standard probability Borel space (X ′ , B ′ , µ ′ ) is a factor of the flow (S t ) t∈R if there exists a measurable map ψ : X → X ′ such that the image of µ via ψ is µ ′ and ψ • S t = S ′ t • ψ for every t ∈ R. Then the flow is (S t ) t∈R called an extension of (S ′ t ) t∈R . If additionally, ψ is finite-to-one almost everywhere then (S t ) t∈R a finite extension of (S ′ t ) t∈R . A measure-preserving flow (S t ) t∈R on a standard probability Borel space (X, B, µ) is rigid if there exists a sequence (t n ), t n → ∞ such that µ(S tn B△B) → 0 as n → ∞ for every B ∈ B.
It is also proved in [11] that a probability measure-preserving flow (S t ) t∈R on (X, B, µ) is mildly mixing iff (S t ) t∈R has no non-trivial rigid factor, i.e. lim inf t→∞ µ(S t B△B) > 0 for every B ∈ B with 0 < µ(B) < 1.
It follows that the mixing property of a flow implies its mild mixing which in turn implies the weak mixing property.
Assume that T is an ergodic automorphism and f : X → R + is in L 1 (X, B, µ). It is well-known (see e.g. [14]) that the special flow T f is weakly mixing if and only if for every s ∈ R \ {0} the equation has no measurable solution ψ : X → S 1 = {z ∈ C : |z| = 1}. Assume moreover that T is rigid, i.e. for some increasing sequence (q n ), µ(T qn A ∩ A) → µ(A) for each A ∈ B. We will make use of the following simple criterion of weak mixing of special flows over rigid systems.
Proposition 2.1. Under the above assumptions suppose additionally that there exists C > 0 such that for every s = 0 and for all n large enough. Then (5) has no measurable solution for s = 0 and therefore the special flow T f is weakly mixing.
Proof. Suppose that for some s = 0 and a measurable ψ : X → S 1 Then for all k ∈ Z \ {0} and all n large enough we have and since clearly ψ k • T qn · ψ k → 1 in measure, when n → ∞, we obtain a contradiction.
We denote by T d the torus R d /Z d which we will constantly identify with the d-cube [0, 1) d . Let λ T d stand for Lebesgue measure on T d .
A homeomorphism T of a compact topological space X is called uniquely ergodic if it admits a unique T -invariant probability Borel measure µ. Then the measurepreserving automorphism T of (X, µ) is ergodic and for every continuous function f : X → C Recall that if T : T d → T d is the rotation by a vector (α 1 , . . . , α d ) ∈ T d such that α 1 , . . . , α d , 1 are independent over Q then T is uniquely ergodic. Moreover, using standard arguments this gives (6) for every Riemann integrable function f : For a real number t denote by {t} its fractional part and by t its distance to the nearest integer number. For an irrational α ∈ T denote by (q n ) its sequence of denominators (see e.g. [17]), that is we have where q 0 = 1, q 1 = a 1 , q n+1 = a n+1 q n + q n−1 p 0 = 0, p 1 = 1, p n+1 = a n+1 p n + p n−1 .
Let [0; a 1 , a 2 , . . . ] stand for the continued fraction expansion of α. The rational numbers p n /q n are called the convergents of the continued fraction. The number α is said to have bounded partial quotients if the sequence (a n ) is bounded. Then there exists a natural number C such that nα ≥ 1/(C|n|) for every non-zero integer n. It follows that q s+1 ≤ Cq s holds for each natural s.

Weak mixing
In this section we will show weak mixing assuming that the roof function f : T 2 → R + is piecewise C 2 and satisfies the von Neumann condition (2) (in the following section we will establish the absence of partial rigidity under the same assumption). We recall that all rotations on tori are rigid. Lemma 3.1 (see [13]). Let h : T → R be a piecewise absolutely continuous map with N discontinuities. Suppose that h ′ : T → R is of bounded variation and Proof. Suppose that 0 ≤ a 1 < . . . < a N < 1 are all discontinuities of h (we set a N +1 = a 1 ). Using integration by parts we obtain (2). Then the special flow T f is weakly mixing.
Proof. Suppose that T 2 f x (x, y) dxdy = 0. The proof of the symmetric case runs similarly. By Proposition 2.1, it suffices to show that there exist C > 0 and n 0 ∈ N such that for every s = 0 and n ≥ n 0 we have T 2 e 2πisf (n) (x,y) dxdy ≤ C/|s|. Since which completes the proof.

Absence of partial rigidity
Let us recall that a flow (S t ) t∈R acting on a standard probability Borel space (2). Then the special flow T f is not partially rigid.
To prove Theorem 4.1 we will need the following.
Then for every t ≥ 2Cn 0 and 0 < ε < c/4 we have Proof. Fix t ≥ 2Cn 0 and 0 < ε < c/4. Notice that, by (8), Notice that for every j ∈ J the function f j is of class C 1 and strictly monotone (because of (9) and (10)) on the interval Since f j is monotone on I i , I i,j is an interval although it can be empty. If I i,j = [z 1 , z 2 ] is not empty then, by (10) and (11), Now suppose that x ∈ I i,j and y ∈ I i,j ′ with j = j ′ . Since x, y are in the same interval of continuity of f j , by (10) and (8), it follows that In particular, there is no overlap between I i,j and I i,j ′ .
. . , s − 1 that are disjoint from intervals I i,j , j ∈ K i and fill up the space between those intervals. In view of (13) and (11) we have |H l | ≥ c/(2jΘ) ≥ c 2 /(4tΘ) for l = 1, . . . , s − 1. Therefore, by (12) and (13), we obtain The proof of the symmetric case runs similarly. Let c, C be positive numbers such that 0 < c ≤ f (x, y) ≤ C for every (x, y) ∈ T 2 . Assume, contrary to our claim, that T f is partially rigid. By Lemma 7.1 in [8], there exist (t n ) n∈N , t n → +∞ and 0 < u ≤ 1 such that for every 0 < ε < c we have Let 0 ≤ a 1 < . . . < a N < 1 and 0 ≤ b 1 < . . . < b M < 1 be points determining the lines of points of discontinuity for f . Since f x : T 2 → R is Riemann integrable, by the unique ergodicity of T , there exist 0 < θ < Θ and m 0 ∈ N such that Now an application of Lemma 4.2 to the sequence (f (m) ( · , y)) m∈N gives whenever t n > 2Cm 0 , contrary to (14).

Weak Ratner's property
In this section we introduce and discuss consequences of weak Ratner's property. Weak Ratner's property will be one more weakening of the classical Ratner condition from [26]. The present idea has already been used in case P is finite in [8] and [9].
Moreover, we say that (S t ) t∈R has the property R(P ) if the set of s ∈ R such that the flow (S t ) t∈R has the R(s, P )-property is uncountable. Flows with the latter property are said to have weak Ratner's property.
Remark 5.1. Note that the original Ratner notion of H p -flow, introduced in [26], is equivalent to requiring that a flow has R(p, {−p, p})-property. The notion we introduce is different from the concept of Ratner's property presented by Witte in [30]. The main difference is that Witte admits compact subsets in the centralizer of the flow (S t ) t∈R as the set of displacements. In our approach this set is included in the flow. It should be emphasized that Witte has used his notion to prove certain rigidity phenomena of some translations on homogeneous space but not to study the structure of joinings which is one of our aims.
The following result is a simple consequence of Birkhoff's Ergodic Theorem.
Lemma 5.2. Let T : (X, B, µ) → (X, B, µ) be an ergodic automorphism and A ∈ B. For every ε > 0, δ > 0 and κ > 0 there exist N = N (ε, δ, κ) ∈ N and X(ε, δ, κ) ∈ B with µ(X(ε, δ, κ)) > 1 − δ such that for every M, L ∈ N with L ≥ N and Remark 5.3. If the set P ⊂ R \ {0} is finite then using Luzin's theorem and Lemma 5.2 one can easily show that the R(s, P )-property does not depend on the choice of the metric d on X compatible with B. We have been unable to decide whether for P infinite (and compact) the R(s, P )-property depends on the choice of the metric; it is very likely that it does. This is why we are forced to put one more assumption on d, see (15) below (see also Remark 5.5 below).
We will constantly assume that (S t ) t∈R satisfies the following "almost continuity" condition for every ε > 0 there exists X(ε) ∈ B with µ(X(ε)) > 1 − ε such that for every ε ′ > 0 there exists ε 1 > 0 such that Notice that if (S t ) t∈R is a special flow acting on a space Y f equipped with a metric of the form (4) then (15) holds. We intend to prove a version of famous Ratner's theorem which describes the structure of ergodic joinings between a system satisfying weak Ratner's property and an arbitrary one, see Theorem 5.9.
Assume that S = (S t ) t∈R and T = (T t ) t∈R are ergodic flows acting on (X, B, µ) and (Y, C, ν) respectively. By a joining one means any (S t × T t ) t∈R -invariant probability measure ρ on (X × Y, B ⊗ C) with the marginals µ and ν respectively. We then write ρ ∈ J(S, T ). The set of ergodic joinings is denoted by J e (S, T ).
An essential step of the proof of Theorem 5.9 will be based on the following result.
Proof of Lemma 5.4 For every ǫ > 0 and p ∈ R set It follows that for every p ∈ R we have and similarly ρ (S −p A × B) − ρ (I(ε 1 , p)) < ε/2.
Let Q ⊂ P be a finite set such that P ⊂ Q + [−ε 1 /2, ε 1 /2]. By Lemma 5.2 applied to T 1 × S 1 : (X × Y, ρ) → (X × Y, ρ) and sets U (ε 1 /2, q), I(ε 1 /2, q) for q ∈ Q, there exist N ∈ N and Θ ⊂ B ⊗ C with ρ(Θ) > 1 − δ such that for every M, L ∈ N with L ≥ N and L/M ≥ κ we have for all (x, y) ∈ Θ and q ∈ Q. Take p ∈ P and choose q ∈ Q such that p ∈ q + [−ε 1 /2, ε 1 /2]. Then which completes the proof. Theorem 5.9. Let (X, d) be a σ-compact metric space, B be the σ-algebra of Borel subsets of X, µ a probability Borel measure on (X, d). Let (S t ) t∈R be a weakly mixing flow on the space (X, B, µ) that satisfies the R(P )-property where P ⊂ R \ {0} is a nonempty compact set. Assume that (S t ) t∈R and (X, d) satisfy (15). Let (T t ) t∈R be an ergodic flow on (Y, C, ν) and let ρ be an ergodic joining of (S t ) t∈R and (T t ) t∈R . Then either ρ = µ ⊗ ν, or ρ is a finite extension of ν.
Proof. Suppose that ρ ∈ J e (S, T ) and ρ = µ ⊗ ν. Since the flow (S t × T t ) t∈R is ergodic on (X × Y, ρ), we can find t 0 = 0 such that the automorphism S t0 × T t0 : (X × Y, ρ) → (X × Y, ρ) is ergodic and the flow (S t ) t∈R has the R(t 0 , P )-property. To simplify notation we assume that t 0 = 1.
By Remark 5.7, there exist two families {A i : i ∈ N} and {B i : i ∈ N} dense in (B, µ) and (C, ν) respectively such that µ(∂A i ) = 0 for all i ∈ N. Let us consider the map and R ∋ t → S t ∈ Aut(X, B, µ) is a continuous representation, the function ̺ is continuous. Notice that ̺(t) > 0 for t = 0. Indeed, if ̺(t) = 0 then ρ(S −t A i ×B j ) = ρ(A i × B j ) for all i, j ∈ N, and hence ρ(S −t A × B) = ρ(A × B) for all A ∈ B, B ∈ C. By the ergodicity of S t , we obtain ρ = µ ⊗ ν.

Now from
Applying similar arguments we get contrary to (21).

Weak Ratner's property for special flows
In this section we present techniques that will help us to prove the weak Ratner property for special flows built over isometries. The following is a general version of Lemma 5.2 in [8]. We omit its proof since it is showed as in [8].
Proposition 6.1. Let (X, d) be a compact metric space, B the σ-algebra of Borel subsets of X and let µ be a probability Borel measure on (X, d). Assume that T : (X, µ) → (X, µ) is an ergodic isometry and f : X → R is a bounded positive measurable function which is bounded away from zero. Let P ⊂ R \ {0} be a nonempty compact subset. Assume that for every ε > 0 and N ∈ N there exist κ = κ(ε) > 0, 0 < δ = δ(ε, N ) < ε and Z = Z(ε, N ) ∈ B, µ(Z) > 1 − ε such that if x, y ∈ Z, 0 < d(x, y) < δ, then there are natural numbers M = M (x, y) ≥ N , L = L(x, y) ≥ N such that L/M ≥ κ and there exists p = p(x, y) ∈ P such that Suppose that γ ∈ R is a positive number such that the γ-time automorphism T f γ : X f → X f is ergodic. Then the special flow T f has the R(γ, P )-property.
is called a-sparse if there exists an increasing sequence (k m ) m≥0 , k 0 = 0, of natural numbers such that (i) x n = 0 with n ≥ 1 if and only if n = k m for some m ≥ 1; Let T : X → X be an isometry of a metric space (X, d). Let f : X → R be a Borel function and let H = {h 1 , . . . , h s }, s ≥ 3, a collection of real numbers. Assume that (23) h 1 , . . . , h s−1 are linearly independent over Q and h s−1 = h s .
Without loss of generality we can assume that 0 The number M will be chosen between C 1 /d and 3sC 2 /d, and we will precise its value later. Then By assumptions, there exists an increasing sequence (k m ) m≥0 , k 0 = 0 of natural numbers such that N j (T n x, T n y) = 0 for k m < n < k m+1 and for all m ≥ 0 and j = 1, . . . , s; (30) for each m ≥ 1 there exists 1 ≤ j ≤ s with N j (T km x, T km y) = 0; We use (34) for m = s + 1 and obtain m 1 , and apply again (34) for m 1 + 1 to have s < m 1 < m 2 ≤ m 1 + s such that k mi+1 − k mi > L for i = 1, 2. It follows that the set is not empty. Pick a pair (m 1 , m 2 ) from this set with the smallest m 2 − m 1 . Then Indeed, suppose contrary to our claim that there exist 1 ≤ j ≤ s and contrary to the definition of (m 1 , m 2 ).

It follows that
In view of (28) and (29), Consequently, which completes the proof.
We will consider now T an isometry of a (compact) metric space (X, d) which is ergodic with respect to a probability Borel measure µ. We will assume that ( X, d) is another metric space. Moreover, we assume that π : ( X, d) → (X, d) is a surjective function which, in addition, is uniformly locally isometric. More precisely, π : B d ( x, 1/2) → B d (π( x), 1/2) is a bijective isometry for every x ∈ X. Let T : X → X be an isometry of ( X, d) such that π • T = T • π. Proof. By Lemma 6.3 applied to T and f , for every 0 < ε < 1/2 and N ∈ N there exist κ = κ(ε) > 0, 0 < δ = δ(ε, N ) < ε such that if x, y ∈ X, 0 < d( x, y) < δ, then there are natural numbers M = M ( x, y) ≥ N , L = L( x, y) ≥ N such that L/M ≥ κ and there exists p = p( x, y) with p 0 ≤ |p| ≤ p 1 such that Let x, y ∈ X arbitrary distinct point such that d(x, y) < δ. By assumption, there are distinct x, y ∈ X such that π( x) = x, π( y) = y and d( x, y) = d(x, y) < δ. Since it follows that T and f verify the assumptions of Proposition 6.1 with P = [−p 1 , −p 0 ]∪ [p 0 , p 1 ]. This gives R(t 0 , P )-property for all t 0 ∈ R \ {0} such that T f t0 is ergodic and weak Ratner's property follows.

Special flows over rotations on the two torus
In this section we will deal with special flows over ergodic rotations T (x, y) = (x + α, y + β) on T 2 . We will constantly assume that both α and β have bounded partial quotients. We will consider roof functions of the form f (x, y) = f 1 (x) + f 2 (y) + g(x, y) + γh(x, y), where f 1 , f 2 : T → R are piecewise C 2 -functions which are not continuous, g : and γ ∈ R. The function h naturally appears when considering rotations on the nil-manifold which is the quotient of the Heisenberg group modulo its subgroup of matrices with integer coefficients.
In order to prove weak Ratner's property for the corresponding special flows, we will apply Proposition 6.4 in which X = T 2 = R 2 /Z 2 , X = R 2 , π : R 2 → T 2 is defined naturally and T is the translation on R 2 by (α, β).
Lemma 7.1. Let α ∈ R be an irrational number with bounded partial quotients. Let us consider the function N : . Then there exist positive constants C 1 , C 2 such that for any pair x, Proof. Since α has bounded partial quotients, there are constants C 1 , C 2 > 0 such that for each m ∈ N the lengths of intervals I in the partition of T arisen from 0, α, . . . , (m − 1)α satisfy Suppose that n 1 < n 2 are natural numbers such that n 1 α, n 2 α ∈ [−x, −x ′ ) + Z and nα / ∈ [−x, −x ′ ) + Z for n 1 < n < n 2 . It follows that the interval [−x, −x ′ ) (as an interval on T) contains exactly one point of the sequence n 1 α, . . . , (n 2 − 1)α, hence Moreover, [−x, −x ′ ) contains exactly two points of the sequence n 1 α, . . . , n 2 α, hence Therefore, which completes the proof. and for distinct x, x ′ ∈ R we have Then, for the translation (x, y) → (x + α, y + β) on R 2 we have It follows that Moreover, Consequently, Theorem 7.4. Let T (x, y) = (x + α, y + β), α, β ∈ [0, 1), be an ergodic rotation on the torus T 2 such that both α and β have bounded partial quotients. Let f : where f 1 , f 2 : T → R are piecewise C 2 -functions which are not continuous and g : by Theorem 3.2, the special flow T f is weakly mixing.
Note that every piecewise C 2 -function F : T → R with s discontinuities ∆ 1 , . . . , ∆ s with jumps d 1 , . . . , d s respectively, can be represented as where F is a continuous function which is piecewise C 2 . Therefore, we can assume that and g is a Lipschitz function.
Since d 1,1 , . . . , d 1,s1 , d 2,1 , . . . , d 2,s2 , γ are independent over Q, the assumptions of Proposition 6.4 are verified with R = 1 and B = |γ|. This completes the proof of weak Ratner's property for the special flow T f in case (i). The proof in case (ii) runs as before.

Mild mixing
Using a result from [8] we will now show mild mixing property for the class of flows from the previous section. Lemma 8.1 (see [8]). Let (S t ) t∈R be an ergodic flow on (X, B, µ) which has finite fibers factor property. Then the flow (S t ) t∈R is mildly mixing provided it is not partially rigid. Proof. In view of Theorem 5.9, since the special flow T f has weak Ratner's property, it is a finite extension of each of its non-trivial factors. As f x (x, y) dxdy = 0 or f y (x, y) dxdy = 0, by Theorem 4.1, T f is not partially rigid. An application of Lemma 8.1 completes the proof.
Example. Let us consider the roof function f : where a, b, c are independent over Q with a = cβ or b = −cα. By Theorem 8.2, the special flow T f is mildly mixing provided that T (x, y) = (x + α, y + β) is an ergodic rotation on the torus T 2 such that both α and β have bounded partial quotients.
In the light of next section it is not however clear whether flows from Theorem 8.2 are not mixing. We will now show that at least some of them are certainly not mixing. The main idea is to find α, β ∈ T so that 1, α, β are rationally independent, α and β have bounded partial quotients and the intersection of the sets of denominators of α and β are infinite. Examples of such α and β have been pointed out to us by M. Keane. Below, we present his argument.
Let (a n ) n≥1 be a palindromic sequence in {1, . . . , N } (for some fixed N ≥ 2), i.e. we assume that (a n ) n≥1 has infinitely many prefixes which are palindromes and (a n ) is not eventually periodic; if in the standard Thue-Morse sequence 01101001 . . . we replace 0 by 1 and 1 by 2 the resulting sequence is palindromic for N = 2, see e.g. [1]. Let It is classical that [0; a 1 , a 2 , . . . , a kn ] = p n q n and [0; a kn , a kn−1 , . . . , a 1 ] = r n q n , so the k n -th denominators of α and β are the same. In this way we have obtained an infinite sequence (q n ) n≥1 for α and β (each q n being the k n -th denominator of α and β). Setting f (x, y) = a{x} + b{y} + c, by the Denjoy-Koksma inequality, |f (qn) (x, y)−q n f dµ| ≤ 2(|a|+|b|). Since (q n ) is a rigidity sequence for the ergodic rotation T (x, y) = (x + α, y + β), by standard arguments (see [18]), the special flow T f is not mixing (in fact, it is not partially mixing, see Section 10).

Mixing
In this section we will show that von Neumann's special flows over ergodic twodimensional rotations can be mixing. We will make use of the following criterion for mixing in which a partial partition of T means a partition of a subset of T. Proposition 9.1 (see Proposition 3.3 in [5]). Let T f be the special flow built over an ergodic rotation T : T 2 → T 2 , T (x, y) = (x + α, y + β) and under a piecewise C 2 roof function f : T 2 → R + . Let (τ n ), (ε n ) and (k n ) be sequences of real positive numbers such that τ n → ∞, ε n → 0, k n → ∞ and let (η n ) be a sequence of partial partitions of T, where η n = {C Suppose that there exists n 0 such that if n ≥ n 0 then • for any m ∈ [τ 2n /2, 2τ 2n+1 ], y ∈ T and C Then T f is mixing.
Remark 9.2. The above criterion for mixing has been formulated by Fayad [5] only for C 2 roof functions. Nevertheless, following word by word Fayad's proof we obtain that the assertion holds whenever f is piecewise C 2 .
As it was observed by Yoccoz in [31, Appendix A] the set of all pairs satisfying (42) is uncountable. Note that the rotation T : T 2 → T 2 , T (x, y) = (x + α, y + β) is ergodic. Indeed, if T is not ergodic then there exist integer numbers k = 0, l = 0 and m such that kα + lβ = m. Next choose n ∈ N such that (43) γ(n) > max(|k|, |l|).
Proof. Let 0 ≤ a 1 < . . . < a N < 1 and 0 ≤ b 1 < . . . < b M < 1 be points determining the lines of discontinuities for f . Since f x , f y : T 2 → R are Riemann integrable function, by the unique ergodicity of T and (3) Choose n 0 ∈ N such that q n0 , r n0 ≥ m 0 . Fix n ≥ n 0 . Let κ stand for the partition (into intervals) of T determined by points a l − jα, 1 ≤ l ≤ N , 0 ≤ j < q n qn+1 γ(n)qn (⌈x⌉ = min{n ∈ Z : x ≤ n}). Set Recall that for every 1 ≤ l ≤ N the diameter of the partition T determined by points a l − jα for 0 ≤ j < q n is bounded by 1 qn + 1 qn+1 . Since η 2n is finer than each such partition, For every pair l, j, where 1 ≤ l ≤ N and 0 ≤ j < q n let us consider the family of points A l,j = a l − (j + iq n )α : 0 ≤ i < q n+1 γ(n)q n .
possible in the class of flows considered in Theorem 7.4? If the answer to the second question is positive then Theorem 7.4 would give the first examples of mixing special flows over rotations having (weak) Ratner's property. For such flows mixing of all order follows; indeed, flows having weak Ratner's property are quasi-simple in the sense of [27] and mixing implies mixing of all orders for such flows [27]. Another possibility to obtain mixing of all orders would be to show that for example if we take f (x, y) = a{x} + b{y} + c and (α, β) satisfying (42) then the spectrum of U T f is singular: mixing of all orders would follow from [12]. We recall that Fayad in [6] has constructed a smooth reparametrization of a linear flow on T 3 which is mixing and has simple singular spectrum. Such a reparametrization flow has a representation as the special flow over a two-dimensional rotation and under a smooth roof function.
Little is known about the spectrum of weak von Neumann's special flows. It seems to be completely open whether such flows can have an absolutely continuous component in the spectrum. This is impossible over rotations on T (in fact, in the one dimensional case we have even spectral disjointness with all mixing flows [7]). It is neither clear whether such flows can have simple spectrum -this remains an open problem even in the one dimensional case.
Finally, it would be nice to decide whether there exists a weak von Neumann's special flow over two-dimensional rotations which is self-similar -this is impossible for von Neumann's special flows over rotations on the circle [10].